#72314045

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623475	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20475 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,4)(3,6,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,28)(20,31)(21,29)(22,32)(23,33)(24,34)(25,35)(26,36)(27,30)(37,39)(38,40);(1,3,5,7)(2,6,4,8)(10,15)(11,19)(17,29)(18,30)(20,37)(22,24)(23,26)(35,40);(1,2,5,4)(3,6,7,8)(11,20,19,37)(12,25)(13,27)(16,31,28,39)(17,35)(18,30)(21,38)(22,23,24,26)(29,40)(32,33,34,36);(2,4)(6,8)(9,11,21,23,13,24,38,37,14,19,12,26,27,22,25,20)(10,16,29,33,18,34,40,39,15,28,17,36,30,32,35,31);(1,4,5,2)(3,6,7,8)(11,22,19,24)(12,21)(16,32,28,34)(17,29)(20,23,37,26)(25,38)(31,33,39,36)(35,40)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=f,d^2=fghj,e^2=f,h^2=i,i^2=j,b^a=bj,d^a=dg,d^b=dj,d^c=dghij,e^b=eg,e^d=egi,f^d=fj,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314251, #72314475
maximal normal subgroup of	#72314098, #72314251, #72314475

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);